[Boromir88]

Some other things that I noticed is, at least at first Saruman probably held more "authority" than Gandalf did...

"The Council of the Wise" being also called the "White Council," because Saruman was the leader. I don't think it's fully known who makes up this Council, but we know Saruman, Gandalf, and Galadriel are definitely in it, for Galadriel wanted to place Gandalf as the "leader." So, this brings to the point that based on this quote:

It was afterwards said that they came out of the Far West and were messengers sent to contest the power of Sauron, and to unite all those who had the will to resist him.

Many understood why the Istari were there. Perhaps they did not know their identity, or knew if they were Maiar. But, they did understand that these men came from the West and were "messengers." And they were sent to Middle-earth for a purpose, or the "Wise" of Middle-earth, would not put them in such high positions and make them leaders:

TA 2851: The White Council meets. Gandalf urges an attack on Dol Guldur. Saruman overrules him...

TA 2941: ...The White Council meets; Saruman agrees to an attack on Dol Guldur, since he now wishes to prevent Sauron from searching the River.

These two dates show the authority that Saruman possesses when it comes to the Council. So, whether or not people knew who exactly these Istari were. I think it was understood that they were sent to help combat Sauron, and which is why they are placed into positions of leadership.

We also get an interesting note on the two blue wizards (who are called Morinehtar and Romestamo in this instance):

The ’other two’ came much earlier, at the same time probably as Glorfindel, when matters became very dangerous in the Second Age Glorfindel was sent to aid Elrond and was (though not yet said) pre-eminent in the war in Eriador.But the other two Istari were sent for a different purpose. Morinehtar and Romestamo. Darkness-slayer and East-helper. Their task was to circumvent Sauron: to bring help to the few tribes of Men that had rebelled from Melkor-worship, to stir -up rebellion...and after his first fall to search out his hiding (in which they failed) and to cause dissension and disarray among the dark East...They must have had very great influence on the history of the Second Age and Third Age in weakening and disarrayinbg the forces of the East...who would both in the Second and Third Age otherwise have ...outnumbered the West.~HoME VIII: The Peoples of Middle-earth, Last Writings (The Blue Wizards)

It seems to be understood that the Istari were sent to aid in fighting Sauron, whether or not it was understood who exactly these "Old men" were.

This suggests that Denethor was aware of the nature of Gandalf's mision (and that of the other Istari) as emissaries of the Valar (and, ultimately, Eru). Is there anything else to suggest that this may have been the case? I was rather under the impression that only a select few (essentially, the White Council and, possibly, Sauron) were aware of this.~Sauce

I don't think it was a secret as to where these Istari came from (generally understood as they were not from Middle-earth), or their "mission," as I have shown. Now whether a majority knew it or not, I don't know, but Gandalf was not shy about professing being emissaries from the "higher authorities":

"I am a servant of the secret fire..."~The Bridge of Khazad-dum

And Gandalf also gives a detailed account of his meeting with Saruman to the Council of Elrond which includes exactly what Saruman felt like their mission was. Plus there's the instance we are given with Denethor, as narfforc shows.